Capital Allocation Between The Risky And The RiskFree

Allocating Capital: Risky & Risk Free Assets It’s possible to split investment funds between safe and risky assets. Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio) 7 -2

Allocating Capital: Risky & Risk Free Assets Issues Examine risk/return tradeoff. Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets. 7 -3

Example Using Chapter 7. 3 Numbers p refers to a risky portfolio, rf is the risk free rf = 7% rf = 0% E(rp) = 15% p = 22% y = % in p (1 -y) = % in rf 7 -4

Expected Returns for Combinations E(rc) = y. E(rp) + (1 - y)rf (Rule 3) rc = complete or combined portfolio For example, y = 0. 75 E(rc) = 0. 75(0. 15) + 0. 25(. 07) = 0. 13 or 13% 7 -5

Portfolio Risk for Combinations Example: let y=0. 75. c=0. 75*0. 22 = 0. 165 Note: Both the expected return formula of the previous slide, and the risk formula for this slide are linear; thus, all combinations will also be linear 7 -6

Risk and Return by y y 1 -y E(r) St. Dev 0. 0 1. 0 0. 070 0. 000 0. 2 0. 8 0. 086 0. 044 0. 6 0. 102 0. 088 0. 6 0. 4 0. 118 0. 132 0. 8 0. 2 0. 134 0. 176 1. 0 0. 150 0. 220 1. 2 -0. 2 0. 166 0. 264 1. 4 -0. 4 0. 182 0. 308 1. 6 -0. 6 0. 198 0. 352 7 -7

Possible Combinations (the CAL) Note: This is a plot of the Capital Allocation Line. Each point on the line is determined by the value of “y” E(r) E(rp) = 15% P E(rc) = 13% C Slope = rf = 7% F 0 (E(rp)- rf)/σp c p=22% 7 -8

Slope of the CAL Slope = (E(rp)- rf)/σp =(0. 15 – 0. 07)/0. 22 =. 08/0. 22 = 0. 3636 This is referred to as the “Reward to Volatility Ratio” 7 -9

Capital Allocation Line with Leverage Borrow at the Risk-Free Rate and invest in stock. Using 50% leverage means that y = 1. 5, so (1 -y) = -0. 5 E(rc ) = (1. 5) (0. 15) + (-0. 5) (0. 07) = 0. 19 c = (1. 5) (0. 22) = 0. 33 7 -10

CAL (Capital Allocation Line) E(r) P E(rp) = 15% E(rp) - rf = 8% ) S = 8/22 rf = 7% F 0 p = 22% 7 -11

CAL with Higher Borrowing Rate E(r) P ) S =. 27 9% 7% ) S =. 36 p = 22% 7 -12

Risk Aversion and Allocation Greater levels of risk aversion lead to larger proportions of the risk free rate. Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets. Willingness to accept high levels of risk for high levels of returns will result in leveraged combinations (i. e. buying the risky asset on margin). 7 -13

Recall: Utility Function U = E (r) ― 0. 005 A 2 Where U = utility E (r) = expected return on the asset or portfolio A = coefficient of risk aversion 2 = variance of returns 7 -14

CAL with Risk Preferences The lender has a larger A when compared to the borrower E(r) P Borrower 7% Lender p = 22% 7 -15

Utility Maximization Max U = E(r) – 0. 005 Aσ2 Can be written as: Max: rf + y[Erp) – rf] – 0. 005 Ay 2σp 2 Take the derivative with respect to y and set to zero to determine the y value that maximized utility to derive: Note: y* depends on A 7 -16

Passive Investment Strategy A passive strategy concentrates on selecting the appropriate asset allocation, and then investing using a buy and hold approach Research shows that most of the return in a portfolio is due to asset allocation rather than individual security selection Passive strategies have very low investment costs 7 -17

Capital Market Line The CAL which uses the risk free asset and the portfolio of all traded assets (usual proxy is the S&P 500) is called the Capital Market Line 1926 -2002 shows an equity risk premium of 8. 2%, and a 20. 8% standard deviation for a reward to variability ratio of 0. 39 Given that overall investors have 74% of their wealth in risk assets (Table 1. 2) the implied risk aversion coefficient is A=2. 6. 7 -18